帶參數時空分數階Fokas-Lenells 方程的精確解*

劉楊秀， 胡彥霞

（華北電力大學 數理學院，北京 102206）

引 言

近幾十年來，分數階非線性偏微分方程的研究已滲透到許多科學領域中.由整數階微分方程推廣出的分數階微分方程，在描述一些系統的動力學行為中，更加能反映出系統的實際變化規律[1].眾所周知的分數階非線性Schr?dinger 方程就是非常典型的一類非線性發展方程.很多分數階非線性偏微分方程與深水波動、潛水波動現象有著緊密的聯系.如果知道這類方程的精確解，將有利于數值模擬進行檢驗以及定性分析.由此我們需要了解分數階導數各種詳細的定義，比如R-L 分數階導數、Caputo 分數階導數等[2-4]，同樣用來解決整數階非線性偏微分方程的方法，也可以應用到分數階非線性偏微分方程上.從現有文獻看，求解分數階非線性偏微分方程的方法已有眾多，比如試探函數法[5]、Horita 法[6]、擴展的直接代數法[7]、李群方法[8-9]、Jacobi 橢圓函數法[10]、擴展的Jacobi 橢圓函數法[11]、廣義G′/G方法[12]、sine-Gordon 方法[13]、多項式完全判別系統法[14]以及其他求解方法[15-24].由于非線性分數階偏微分方程的多樣性及復雜性和每一種求解方法的局限性，還未能有一種通用的普遍有效的求解方法.因此，運用有效的方法去求非線性分數階偏微分方程的精確解仍是一個需要不斷研究的課題之一.隨著MATLAB 等其他數學軟件的應用和普及，借助計算機軟件可以幫助我們更方便地解決這一問題.

本文主要考慮非線性光學領域中的帶參數時空分數階Fokas-Lenells 方程[25]：

多項式完全判別系統法是一種求解此類問題比較有效的方法，是將偏微分方程在行波變換下簡化為常微分方程，再對常微分方程中的多項式進行完整分類并求解相應積分，從而得到原方程的精確解.對于分數階偏微分方程，若其在行波變換下簡化為u′(ξ)=G(u,θ1,θ2,···,θm)(θ1,θ2,···,θm為 相應參數，且G(u,θ1,θ2,···,θm)是關于u的多項式)的形式，就可以利用此方法進行求解.關于此方法的介紹及應用詳見文獻[37-41].

本文考慮方程(1)在一般情況下的解的問題，利用多項式完全判別系統法，根據對方程(1)單行波解的完整分類，在不做任何參數限制的條件下，求得方程(1)在一般情況下的9 種精確解，包括有理函數解、周期解、孤波解、Jacobi 橢圓函數解和雙曲函數解等，繪制了精確解的相關圖像，由此分析了參數對解的結構的影響.

1 預 備 知 識

2 帶參數時空分數階Fokas-Lenells 方程的約化分析

首先對方程進行行波變換，令

3 帶參數Fokas-Lenells 方程的精確解

圖1 | Ω1(x,t)|2取 不同分數階導數值時的三維圖，圖1(b)對應的等高線圖，以及t =3 時| Ω1(x,t)|2關 于 x 的 截面圖：(a) α =1/2,β=1/3,|Ω1(x,t)|2的三維圖； (b) α =1/3,β=1/3,|Ω1(x,t)|2 的 三維圖； (c) 圖1(b)的等高線圖； (d) 當t =3 時| Ω1(x,t)|2關 于x 的截面圖Fig. 1 The 3D graph of | Ω1(x,t)|2 with different values of the fractional derivative, the contour plot of fig.1(b) and the sectional view of | Ω1(x,t)|2 against x with t =3: (a) the graphic model of | Ω1(x,t)|2,α=1/2,β=1/3; (b) the graphic model of | Ω1(x,t)|2,α=1/3,β=1/3; (c) the contour plot of fig.1(b);(d) the sectional view of | Ω1(x,t)|2 against x when t=3

圖2 | Ω2(x,t)|2取 不同分數階導數值時的三維圖，圖2(b)對應的等高線圖，以及t =3 時| Ω2(x,t)|2關 于 x 的 截面圖：(a) α =1/2,β=1/3,|Ω2(x,t)|2的三維圖； (b) α =1/3,β=1/3,|Ω2(x,t)|2 的 三維圖； (c) 圖2(b)的等高線圖； (d) 當t =3 時| Ω2(x,t)|2關 于x 的截面圖Fig. 2 The 3D graph of | Ω2(x,t)|2 with different values of the fractional derivative, the contour plot of fig.2(b) and the sectional view of | Ω2(x,t)|2 against x with t =3: (a) the graphic model of | Ω2(x,t)|2,α=1/2,β=1/3; (b) the graphic model of | Ω2(x,t)|2,α=1/3,β=1/3; (c) the contour plot of fig.2(b);(d) the sectional view of | Ω2(x,t)|2 against x when t=3

經檢驗，此情況不成立，不予討論.

圖3 | Ω3(x,t)|2取 不同分數階導數值時的三維圖，圖3(b)對應的等高線圖，以及t =3 時| Ω3(x,t)|2關 于 x 的 截面圖：(a) α =1/2,β=1/3,|Ω3(x,t)|2的三維圖； (b) α =1/3,β=1/3,|Ω3(x,t)|2 的 三維圖； (c) 圖3(b)的等高線圖； (d) 當t =3 時| Ω3(x,t)|2關 于x 的截面圖Fig. 3 The 3D graph of | Ω3(x,t)|2 with different values of the fractional derivative, the contour map of fig. 3(b) and the sectional view of | Ω3(x,t)|2 against x with t =3: (a) the graphic model of | Ω3(x,t)|2,α=1/2,β=1/3; (b) the graphic model of | Ω3(x,t)|2,α=1/3,β=1/3; (c) the contour plot of fig. 3(b);(d) the sectional view of | Ω3(x,t)|2 against x when t=3

圖4 | Ω4(x,t)|2取 不同分數階導數值時的三維圖，圖4(b)對應的等高線圖，以及t =3 時| Ω4(x,t)|2關 于 x 的 截面圖：(a) α =1/2,β=1/3,|Ω4(x,t)|2的三維圖； (b) α =1/3,β=1/3,|Ω4(x,t)|2 的 三維圖； (c) 圖4(b)的等高線圖； (d) 當t =3 時| Ω4(x,t)|2關 于x 的截面圖Fig. 4 The 3D graph of | Ω4(x,t)|2 with different values of the fractional derivative, the contour plot of fig.4(b) and the sectional view of | Ω4(x,t)|2 against x with t =3: (a) the graphic model of | Ω4(x,t)|2,α=1/2,β=1/3; (b) the graphic model of | Ω4(x,t)|2,α=1/3,β=1/3; (c) the contour plot of fig.4(b);(d) the sectional view of | Ω4(x,t)|2 against x when t=3

圖5 | Ω5(x,t)|2取 不同分數階導數值時的三維圖，圖5(b)對應的等高線圖，以及t =3 時| Ω5(x,t)|2關 于 x 的 截面圖：(a) α =1/2,β=1/3,|Ω5(x,t)|2的三維圖； (b) α =1/3,β=1/3,|Ω5(x,t)|2 的 三維圖； (c) 圖5(b)的等高線圖； (d) 當t =3 時| Ω5(x,t)|2關 于x 的截面圖Fig. 5 The 3D graph of | Ω5(x,t)|2 with different values of the fractional derivative, the contour plot of fig. 5(b) and the sectional view of | Ω5(x,t)|2 against x with t =3: (a) the graphic model of | Ω5(x,t)|2,α=1/2,β=1/3; (b) the graphic model of | Ω5(x,t)|2,α=1/3,β=1/3; (c) the contour plot of fig. 5(b);(d) the sectional view of | Ω5(x,t)|2 against x when t=3

圖6 | Ω6(x,t)|2取 不同分數階導數值時的三維圖，圖6(b)對應的等高線圖，以及t =3 時| Ω6(x,t)|2關 于 x 的 截面圖：(a) α =1/2,β=1/3,|Ω6(x,t)|2的三維圖； (b) α =1/3,β=1/3,|Ω6(x,t)|2 的 三維圖； (c) 圖6(b)的等高線圖；(d) 當t =3 時| Ω6(x,t)|2關 于x 的截面圖Fig. 6 The 3D graph of | Ω6(x,t)|2 with different values of the fractional derivative, the contour plot of fig. 6(b) and the sectional view of | Ω6(x,t)|2 against x with t =3: (a) the graphic model of | Ω6(x,t)|2,α=1/2,β=1/3; (b) the graphic model of | Ω6(x,t)|2,α=1/3,β=1/3; (c) the contour plot of fig. 6(b);(d) the sectional view of | Ω6(x,t)|2 against x when t=3

圖7 | Ω7(x,t)|2取 不同分數階導數值時的三維圖，圖7(b)對應的等高線圖，以及t =3 時| Ω7(x,t)|2關 于 x 的 截面圖：(a) α =1/2,β=1/3,|Ω7(x,t)|2的三維圖； (b) α =1/3,β=1/3,|Ω7(x,t)|2 的 三維圖； (c) 圖7(b)的等高線圖； (d) 當t =3 時| Ω7(x,t)|2關 于x 的截面圖Fig. 7 The 3D graph of | Ω7(x,t)|2 with different values of the fractional derivative, the contour plot of fig.7(b) and the sectional view of | Ω7(x,t)|2 against x with t =3: (a) the graphic model of | Ω7(x,t)|2,α=1/2,β=1/3; (b) the graphic model of | Ω7(x,t)|2,α=1/3,β=1/3; (c) the contour plot of fig.7(b); (d) the sectional view of | Ω7(x,t)|2 against x when t=3

圖8 | Ω8(x,t)|2取 不同分數階導數值時的三維圖，圖8(b)對應的等高線圖，以及t =3 時| Ω8(x,t)|2關 于 x 的 截面圖：(a) α =1/2,β=1/3,|Ω8(x,t)|2的三維圖； (b) α =1/3,β=1/3,|Ω8(x,t)|2 的 三維圖； (c) 圖8(b)的等高線圖； (d) 當t =3 時α =1/2,β=1/3,|Ω8(x,t)|2 關 于x 的截面圖Fig. 8 The 3D graph of | Ω8(x,t)|2 with different values of the fractional derivative, the contour plot of fig. 8(b) and the sectional view of | Ω8(x,t)|2 against x with t =3: (a) the graphic model of | Ω8(x,t)|2,α=1/2,β=1/3; (b) the graphic model of |Ω 8(x,t)|2,α=1/3,β=1/3; (c) the contour plot of fig. 8(b);(d) the sectional view of | Ω8(x,t)|2 against x when t=3

圖9 | Ω9(x,t)|2取 不同分數階導數值時的三維圖，圖9(b)對應的等高線圖，以及t =3 時| Ω9(x,t)|2關 于 x 的 截面圖：(a) α =1/2,β=1/3,|Ω9(x,t)|2的三維圖； (b) α =1/3,β=1/3,|Ω9(x,t)|2 的 三維圖； (c) 圖9(b)的等高線圖； (d) 當t =3 時| Ω9(x,t)|2關 于x 的截面圖Fig. 9 The 3D graph of | Ω9(x,t)|2 with different values of the fractional derivative, the contour plot of fig.9(b) and the sectional view of | Ω9(x,t)|2 against x with t =3: (a) the graphic model of | Ω9(x,t)|2,α=1/2,β=1/3; (b) the graphic model of | Ω9(x,t)|2,α=1/3,β=1/3; (c) the contour plot of fig.9(b);(d) the sectional view of | Ω9(x,t)|2 against x when t=3

4 結 果 分 析

方程精確解的結構反映了光學系統描述的波在介質中傳播的特性.這里，我們主要討論分析分數階參數α ， β 的變化對解的結構的影響.根據前面對方程(1)精確解的 |Ωi(x,t)|2的相關圖形的分析發現，在圖1、3、5、7、9 中，當其中一參數 β不變，參數α 變化時，對應的方程的奇異解的結構沒有發生本質的變化，只有波峰會向左或向右偏移，或出現幾個零散的波峰.由于這些奇異解在波峰處有一個不連續的一階導數，這反映出對應的方程(1)描述的波傳播的特性.在圖2、8 中，當參數α 變化時，奇異周期解的結構及奇異性、周期性也并未發生大的改變.在圖4 中，隨著 α的減小，扭波峰值會向右移動，但扭波仍然存在，其等高線分布僅在很小的時間范圍內波動，但在t=0.1左 右出現新的走勢.在圖6 中，隨著α 的變化，對應的奇異解的結構也沒有發生質的變化，但從等高線密集程度來看，有幾條明顯的“山脈”，反映了此時方程(1)描述的波傳播的情況.

5 結 論

本文通過行波變換，將光學系統中帶參數時空分數階Fokas-Lenells 方程轉換成常微分方程，然后利用多項式完全判別系統法對該方程進行了單行波解的完整分類，在不對方程中的參數和n做任何限定的情況下，得到了方程在一般情況下的精確解，包括有理函數解、孤立波解、雙曲函數解、周期解、Jacobi 橢圓函數解等.在現有文獻中，還沒有見到對帶參數時空分數階Fokas-Lenells 方程中的參數和n不做任何限定的情況下求的精確解的相關結論.這是本文與其他已有文獻不同的地方.為了更好地理解此模型的物理現象和研究光孤子的傳播特性，我們繪制了精確解的相關三維圖、等高線圖及截面圖，討論了分數階參數對解的影響.隨著分數階導數值的變化，比如 β 不變， α減小時，方程的解的結構并未發生質的變化，這也反映出了此時光脈沖在介質中的傳播特性.多項式完全判別系統法不僅可以用來求偏微分方程的精確解，也可以對方程進行定性分析，這也將是我們此后工作的一部分.

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